Population size estimation via alternative parametrizations for Poisson mixture models

We exploit a suitable moment-based reparametrization of the Poisson mixtures distributions for developing classical and Bayesian inference for the unknown size of a finite population in the presence of count data. Here we put particular emphasis on suitable mappings between ordinary moments and recurrence coefficients that will allow us to implement standard maximization routines and MCMC routines in a more convenient parameter space. We assess the comparative performance of our approach in real data applications and in a simulation study.